Euthyphro

Euthyphro (Greek Ευθύφρων) is the name of one of Plato's early Socratic dialogues. Together with three other of his dialogues - the Apology, Crito and Phaedo, it forms part of a group of Platonic dialogues on the trial and death of Socrates. In modern times, the group is often published together in a single volume with the title The Last Days of Socrates.

In Euthyphro, Socrates carries on a discussion on the nature of piety with Euthyphro, an acquaintance whom he meets on the steps of the courthouse as he (Socrates) arrives to answer to charges of impiety brought against him. Socrates discovers that Euthyphro is there in the capacity of a litigant bringing charges against his own father over the death of a laborer in his employ, one who had been retained because of his involvement in the death of a slave. For pressing charges against his father, Euthyphro had been reproached for being impious and it was this which led Socrates to inquiry about the nature of piety.

The problem of Euthyphro

"The point which I should first wish to understand is whether the pious or holy is beloved by the gods because it is holy, or holy because it is beloved by the gods."

Implications of the Problem

In this context, piety and holiness are interchangable with virtue or goodness.
Socrates is asking Euthyphro whether: (a) the gods love something because it is good, or (b) whether the object is good because the gods love it.
If we take (a) to be true, then the gods are unnecessary, because the object would be good regardless of their existence.
If we take (b) to be true, then we have two problems: that of continuity and that of arbitrariness. The problem of continuity arises when the gods change their minds and then declare a new moral law that is substantially different from, or perhaps even the opposite of, the previous moral law. An exaggerated example would be if murder or genocide were declared holy and just or even obligatory. The problem of arbitrariness states that the gods just selected some random things and then declared them holy for no real moral reason (as morality wouldn't exist until they did); this obviously raises the question about whether the random things they selected to be holy were really the best things for humanity.